DailyDirt: Learning How To Do Math Like A Boss
from the urls-we-dig-up dept
Some people claim that they are not "math people" -- that their brains just don't understand mathematics that way "normal" people are supposed to learn it. Perhaps that's true for some, but the subject of math seems to be taught in a way that tends to weed people out as concepts get more abstract. Educators are trying to figure out how to avoid making math lessons as painful as they might have been in the past (and hopefully not create any further torture with "new math" or even "newer new math"). Here are just a few links on changing the way these skills are taught.- Math classes are usually taught as a progression of skills, starting with counting, then adding/subtraction, then multiplication/division, then algebra, maybe some geometry/trig, and ending in calculus. However, a growing trend of abolishing this artificial math progression could have toddlers learning calculus concepts -- and other "advanced" math topics -- in a non-linear order. [url]
- Psychologists see evidence that preschoolers have a natural ability to do basic algebra before they ever sit in a classroom. This ability seems to come from kids' "Approximate Number System" (ANS) or "number sense" -- which allows children to solve simple algebra problems without a systematic logic. Researchers would like to figure out how this number sense breaks down as kids go to school and appear to lose these inherent abilities. [url]
- There's a trend to try to teach *everyone* how to program, but maybe if we're really serious about doing that, we should change the way programming is done. Forcing humans to think like computers isn't going to make programming very popular, so making human-friendly programming languages might be the way to go. [url]
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Filed Under: algebra, ans, approximate number system, calculus, coding, education, intuition, learning, math, number sense, preschoolers, programming, stem, teachers, teaching
Reader Comments
The First Word
“This post on reddit seemed like a good example of new math
http://www.reddit.com/r/explainlikeimfive/comments/205tu9/eli5_this_math_homework_for_my_first_grade r/Spoiler: it turns out the problems were confounded by some obscure PDF glitch.
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What Exactly Is New Here?
The list of “human-friendly” programming languages is longer than a gorilla’s arm: COBOL, BASIC, LOGO, SmallTalk, HyperCard/HyperScript... Seems like every single person who has wondered why programming is so hard falls into the same trap of thinking, if only programming languages were easier to use, more approachable to people with no programming ability, that would actually make a difference.
It doesn’t.
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Re: What Exactly Is New Here?
All that complexity that people find difficult exists for a reason, and it doesn't go away just because someone builds a new language; the complexity has just been swept under the rug by the new language, which is great for really simple tasks--the kind that language designers write tiny little demos for to show off how awesome their new programming language is--but when they try to do anything non-trivial, complexity rears its ugly head surprisingly fast. And that's when people who don't know how to think like a computer end up creating bug-ridden monstrosities that break if you look at them funny.
Anything that weeds them out before they even get started is good, for me and for anyone who uses a computer that doesn't end up with their code loaded onto it.
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Re: What Exactly Is New Here?
This is done in order to sell more textbooks. There is nothing new except the cost.
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Re: This is done in order to sell more textbooks.
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epic games
No.
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Math is hard -
Computer programming, being in many ways a form of applied mathematics, requires the same type of discipline and precision. At some point, once one moves beyond the relatively trivial cases the required discipline and precision makes programming difficult and tedious.
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Re: Math is hard -
I can totally believe that. I'm a really good programmer, but I have zero artistic skill, no matter how I try to develop some. It's obvious that there are certain things that some people just have a knack for and others don't; but in a few fields, that knack is just necessary to even get started. I think programming is one of them, due to "the required discipline and precision" mentioned above.
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Re: Re: Math is hard -
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Re: Re: Re: Math is hard -
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Re: Re: Re: Re: Math is hard -
*maybe* english has the advantage here in that it is so convoluted that to communicate precisely in this unholy monstrosity requires attention to nuance and details at levels higher than typical in other languages, but I think that's a stretch.
the simple fact is that the hardest parts of computer programming are identifying the problem accurately and envisioning a rule based process that actually solves the problem. everything else is mere technical expertise.
there are "higher orders" of good programming focusing on aspects of the code (reusability, composability, flexibility, robustness, abstractness, etc) but if you don't have those two core competencies above then all those other things don't matter. it should also be noted that many of the problems with "business" or "enterprise" code arises because either the business problem or the process is poorly defined in the minds of those running the business (not even getting into rapid changes). and it's not exclusively "either-or" - often it's both. then tack on to that the difficulty in expressing these business concerns accurately to the developers....
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Re: Re: Re: Re: Re: Math is hard -
Well said.
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20 A=1 to 500:NEXT A
30 GOSUB 50
40 GO TO 10
50 PRINT "IT HELPED ME LEARN VARIABLES!":RETURN
RUN
I QUITE LIKED BASIC
IT HELPED ME LEARN VARIABLES!
I QUITE LIKED BASIC
I QUITE LIKED BASIC
I QUITE LI
?ERROR
BREAK IN 10
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I never learned calculus...I did everything else, but never calculus.
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Math is not that useful
Like, the grade school stuff (eg learning to count; how to add, subtract, multiply and divide; what fractions are; what percentages are; what a decimal point is) is still stuff that I use every day.
I've only very rarely had to work out an algebra "solve for X" style problem in my real life, and I have pretty much never needed any of the higher level stuff.
I memorized the quadratic formula, but to this day not only have I personally never had to use it, I couldn't even tell you who would have to use it or what they would use it for.
I went to college late, and by the time I got into film school (Where some knowledge of Trig would've given me more appreciation of things like waveforms) I'd completely forgotten all of Trig.
I was good at math, and still enjoy it, but the way it was taught was so completely and utterly divorced from the actual real world that it never stuck in my head. I learned the quadratic formula not because knowing it would actually be a useful skill, but because some distant authority figure forced me to for reasons they never bothered to explain.
That's not a good way to learn.
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Maths Is Extremely Useful
When I was in my teens, number theory was considered “pure” maths (not having any practical application). But today it is at the very core of strong cryptographic algorithms that we rely on for the security of our Internet communications every day. Do they call it “applied” maths yet? More likely, nobody bothers about the distinction between “pure” and “applied” any more...
*The “s” in “maths” is not silent.
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Re: Maths Is Extremely Useful
I did not, which is really my complaint. The way I and many others were taught Math (We only get one in the US) was completely abstracted from any meaning.
This is tangential to the question of how to teach math at a young age, but it would make me happy if there were some effort to show some applications of math in the classrooms of older students. If complex numbers are a good way to work with waveforms, why not teach them by working with waveforms?
Actually, in general, school is very much "You will learn this subject because we say you will". In my public school days there really was never any attempt at applied Math at all; you learned the quadratic formula because they wouldn't graduate you unless you did, and that was the only reason.
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Re: Maths Is Extremely Useful
It is in the US, since there is no "s" in math in the US.
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Re: there is no "s" in math in the US.
Coincidence?
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Math Programming
We are embedded in societies where communication is mathematically illiterate. Just look at textual posts like this with textual comments.
(((General Communication) contains Maths) && ((Keyboard / Character-sets) contain (Maths Symbols)) → (We'd be able to use it more)
I was unable to write the above using decent symbols as they aren't available. In other words, the way our so called civilisation has "developed" is intrinsically hostile to maths. (The dinosaur brains have effectively bred more of themselves by preventing sensible expression of thought...)
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No it doesn't. Differential equations and linear algebra are very useful in many fields.
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Testing some entities, to achieve mathematical expression
∀ ∃ ∪ ⊆ ∫
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Re: Testing some entities, to achieve mathematical expression
Δ Optimism > 0
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Where Do You Find A Teacher.
http://rowboats-sd-ca.com/adtodd1a/math_ed.htm
Below a certain age level, mathematics is taught by people who are profoundly un-mathematical, and all kinds of theoretical programs-- such as New Math-- grind to a halt upon this fact. Children are nothing if not logical, and they have the single-minded-ness of limited experience. Under the right circumstances, they can learn quite a lot of mathematics, computer programming, etc., but the difficulty is to create those circumstances.
My query to anyone with a proposed system is, do you know how to write it up as a computer program? If yes, then you can always publish it like a book.
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This post on reddit seemed like a good example of new math
Spoiler: it turns out the problems were confounded by some obscure PDF glitch.
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Programming
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Re: Programming
thought(s) != instruction(s)
This may be a difficult concept to grasp, but the two are simply not equatable, translatable or even remotely similar.
I know hollywood likes to make movies where robots with artificial intelligence take over the world because they are far superior than their human creators, but this is pure hollywood bunk. The machines may eventually eliminate humans but it will not be because they follow instructions which give them creative thought(s) and intelligence.
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Noise and fury...
Perhaps it's a difference in volume of study, but I see programming (my art, craft, and trade since 1986) as applied mathematics - Boolean algebra. I've worked with hundreds of programmers over the course of nearly three decades of professional experience. Those who were math-illiterate were on average less capable than those with degree work that included extensive training in math topics. Yes, your experience may vary - you personally may be god's gift to programming without the capacity to recall the quadratic formula, but that makes you an unlikely event in the universe of my experience; it also doesn't mean you're not doing math - simply that you're unaware of the fact...programs are proofs ( http://en.wikipedia.org/wiki/Curry-Howard_correspondence ).
Math SHOULD be the easiest thing to teach/learn due to its perfect consistency, i.e., lack of exceptions. However, we have shrouded the discipline with elitist arguments that excuse teachers when they perform poorly and permit the ill-educated to pretend they are not deficient in adult skills. Mathiness is not genetic beyond the sense of "genetic" meaning "human." Whether an individual student is more predisposed to learning by intensional methods or extensional ones, the same concepts CAN be imparted. The failing of our schools in the main part is the attempt to enforce that ALL students learn almost exclusively by intension since it's faster...when it works. We aren't very good yet at tailoring education for individual students. We mass-produce an inferior product since it has historically been deemed mostly, sort of, good enough.
When I hear claims like "I've never used it," "I can't imagine what good it is," etc., I recognize victims of our pathetic education system, their intellectual lives less rich than mine for having been short-changed by our schools.
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Re: I see programming ... as applied mathematics
There is a difference: in mathematics in you have computability theory, which is deals with the question of whether something is computable or not. But in actual programming, you are worried about efficiency, which means, can you get the computation done with the available resources (time, storage).
For example, inverting a cryptographic hash is, mathematically speaking, a computable problem, but in practice it would take longer than the age of the observable Universe.
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Re: Re: I see programming ... as applied mathematics
Yes. Yes, it is.
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Re: Yes, it is.
And yet they don’t.
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Re: Re: Yes, it is.
I compared notes with my mother, who is a pure mathematician, and found that I knew more about differential equations than she did, but less about things like set theory and number theory.
There are two kinds of physics, classical physics and modern physics. Physics departments teach mostly modern physics, the physics of extremes: very hot, very cold, very fast, very small. In engineering school, there is a sequence of courses, about a year's work, called Engineering Fundamentals. This amounts to a bachelor's degree in classical physics, the kind used by Civil and Mechanical Engineers, dealing with a lot of topics which are beneath a pure physicist's dignity, eg. Elasticity ("Strength of Materials"), Physical Chemistry (Nature and Properties of Materials, or NAPOM), etc. Even where subjects are covered in both branches, they are very different. Engineering thermodynamics, that is, classical thermodynamics, tends to be concerned with "change of state," that is, boiling or freezing, and with heat transfer in viscous flow, wheres modern thermodynamics tends to be concerned with the thermodynamic behavior of ionized particles in a vacuum.
There aren't very many "pure" electrical engineers, any more, that is, electrical engineer's concerned with the use of physical phenomena. Back in Jack Kilby's day, physics was considered a good entree to electrical engineering, but that was back when people were still inventing different kinds of transistors, lasers, etc., and getting Nobel Prizes in Physics for them. There are still some Radio Engineers, who are concerned with signal propagation. However, the solid mass of electrical engineers at this point seem to be de-facto computer programmers, who really don't do physics anymore.
I don't know very much about Civil Engineering, but I gather their big applied physics subject is something called "Soils," that is, the physics of colloids, fluids with large quantities of solid matter suspended in them. To put it another way, the physics of a mudslide. Then, too, Civil Engineering is cognate to Architecture. Above a certain level, a civil engineer has to be an architect as well, and a good architect at that.
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Take a global view
The reverse may be true. The way computer thinks is the universal way. We humans, normally, think in our individual way, and forgot to see the universal approaches for everything.
Microprocessor is a miracle in that sense. Give any problem to it, it will solve all of them in the same way. It has digital registries, core level arithmetic-logic unit, serial parallel converters, and memory update techniques. Thus there is an underlying common idea behind all problems, which engineers have converted into digital hardware, that allows all problems to be solved in only one way.
So, think digitally, using core assembly language, for microprocessor, not PC, then everything will appear same to you. Take a look at https://theoryofsouls.wordpress.com/ for common or global views for everything.
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